System and method for single-carrier space-frequency block coded transmission over frequency selective and fast fading channels

ABSTRACT

A single carrier transmission scheme which utilizes space-frequency block coding and frequency domain equalization (SF-SCFDE) is proposed for frequency selective and fast fading channel. It is shown that employing this technique in slow fading environment depicts the same performance as that obtained with space-time coding scheme. However, in the more difficult fast fading channels, the proposed scheme exhibits much better performance.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

The present application relates to U.S. Provisional Patent Application60/874,144, filed on Dec. 11, 2006, and entitled “METHOD AND SYSTEM FORA SINGLE CARRIER SPACE-FREQUENCY BLOCK CODED TRANSMISSION OVER FREQUENCYSELECTIVE AND FAST FADING CHANNELS,” which is incorporated herein in itsentirety and forms a basis for a claim of priority.

FIELD

Embodiments of the present application relate to the field of singlecarrier transmission schemes in communication systems. Exemplaryembodiments relate to a method and system for single carrierspace-frequency block coding transmission.

BACKGROUND

The rapid increase of the demand for wireless applications hasstimulated tremendous research efforts in developing systems thatsupport reliable high rate transmissions over wireless channels.However, these developments must cope with challenges such as multipathfading, time varying nature of the wireless channel, bandwidthrestrictions and handheld devices power limitations. Space-timetransmission techniques have been proven to combat the detrimentalefforts of the multipath fading. Unfortunately, the large delay spreadsof frequency selective fading channels destroy the orthogonality of thereceived signals which is critical for space-time coding. Consequently,the techniques are only effective over frequency flat block fadingchannels. Orthogonal frequency division multiplexing (OFDM) has alsoshown to combat the multipath fading. A space-time OFDM (ST-OFDM) andspace-frequency OFDM (SF-OFDM) have been proposed as an effective way tocombat the frequency selectivity of the channel. Moreover, SF-OFDM canbe applied to fast fading channel wherein the channel doesn't need to beconstant for at least two block transmission as it is usually requiredfor ST-OFDM scheme. OFDM is a multicarrier communication technique, withwhich a single data stream is transmitted over a number of lower ratesubcarriers. A multicarrier signal consists of a number of independentmodulated subcarriers that can cause a large peak-to-average PAPR whenthe subcarriers are added up coherently. Also OFDM suffer from phasenoice and the frequency offset problems. Therefore, to combat thefrequency selectivity of the channel, an alternative solution for OFDMwas proposed that utilizes single carrier transmission with frequencydomain equalization. In parallel to ST-OFDM scheme, a space-time singlecarrier (ST-SC) transmission scheme was proposed in that requires thechannel to be same for at least two block periods.

Although many high rate wireless communication method and systems havebeen proposed, none provide a space-frequency single carrier (SF-SC)technique which doesn't require the channel to be the same for two blockperiods, and hence beneficial for fast fading channel. Prior methods andsystems do not use a single carrier transmission technique thatimplements space-frequency block coding with additional frequencydiversity as shown in the next section.

SUMMARY

Aspects of the exemplary embodiments are directed to a single carriertransmission scheme which utilizes space-frequency block coding andfrequency domain equalization (SF-SCFDE). Such a technique can be usedwith frequency selective and fast fading channel.

In one exemplary embodiment, a method for single carrier space-frequency(SF-SCFDE) transmission over frequency selective and fast fading channelincludes receiving communication block streams, encoding the receivedcommunication block streams to produce communication blocks, adding acyclic prefix to each communication block to form transmission blocks,and communicating each transmission block through a frequency selectivefading channel.

In another exemplary embodiment, an apparatus for single carrierspace-frequency (SF-SCFDE) transmission over frequency selective andfast fading channel includes a space-frequency encoder that receives andencodes communication block streams, a cyclic prefix adder that adds acyclic prefix to each communication block, and antennas that communicateeach communication block.

In yet another exemplary embodiment, a system for single carrierspace-frequency (SF-SCFDE) transmission over frequency selective andfast fading channel includes a transmitter and a receiver. Thetransmitter includes a space-frequency encoder that receives and encodescommunication block streams, a cyclic prefix adder that adds a cyclicprefix to each communication block, and antennas that communicate eachcommunication block. The receiver includes antennas that receivecommunication blocks from the transmitter, a structure to remove thecyclic prefix, and a decoder.

These and other features, aspects and advantages of the presentinvention will become apparent from the following description, appendedclaims, and the accompanying exemplary embodiments shown in thedrawings, which are briefly described below.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a general diagram of a transmitter in accordance with anexemplary embodiment.

FIG. 2 is a flow diagram depicting operations performed in the singlecarrier space-frequency (SF-SCFDE) transmission scheme of thetransmitter of FIG. 1.

FIG. 3 is a graph comparing the single carrier space-frequency(SF-SCFDE) transmission scheme of the transmitter of FIG. 1 with othertransmission schemes.

DETAILED DESCRIPTION

Exemplary embodiments are described below with reference to theaccompanying drawings. It should be understood that the followingdescription is intended to describe exemplary embodiments of theinvention, and not to limit the invention.

FIG. 1 illustrates a transmitter 10 having N_(t)=2 transmit antennas andN_(t)=1 receive antenna. The information bearing data symbols d(n)belonging to an alphabet A are first demultiplexed into two blockstreams of N symbols each for transmission for either transmit antennasusing a demultiplexer 12. The communication block is represented bys_(k)=[s_(k)(0), s_(k)(1), . . . s_(k)(N−1]^(T) and is transmitted bythe k^(th) transmit antenna, k=1.2 with |s_(k)(n)|²=1, whose N-pointDiscrete Fourier Transform (DFT) is given by |S_(k)=(0), S_(k)(1), . . ., S_(k)(N−1)|^(T). The two symbols blocks s₁ and s₂ are then fed to aspace-frequency encoder 14 suitable for SCFDE to produce the followingtwo blocks:

u ₁ =[s _(1,0) ,−s* _(2,0) ,s _(1,1) ,−s* _(2,N−1) , . . . , s _(1,N−1),−s* _(2,1)]^(T)

u ₂ =[s _(2,0) ,s* _(1,0) ,s _(2,1) ,s* _(1,N−1) , . . . , s _(2,N−1),s* _(1,1)]^(T)  (1)

where s_(k,n)=s_(k)(n). These two blocks u₁ and u₂ are then compressedfrom symbol duration T_(s) to T_(s)/2 and the compressed vector isrepeated twice to form the following blocks

v _(k) =[u _(k) ^(T) ,u _(k) ^(T)]^(T); k=1,2  (2)

The vector v_(k) is denoted by v_(k,e) (v_(k,o)) with the odd (even)elements made zeros i.e. v_(k,e)=[s_(k,e) ^(T),s_(k,e) ^(T)]^(T), k=1, 2and v_(1,o)[−s_(2,o) ^(T),−s_(2,o) ^(T)]^(T), v_(2,o)=[s_(1,o)^(T),s_(1,o) ^(T)]^(T) where

s _(k,e) =[s _(k)(0),0,s _(k)(1),0, . . . , s ₁(N−1),0]^(T)

s _(k,o)=[0,s* _(k)(0),0,s* _(k)(N−1), . . . , 0,s* _(k)(2),0,s*_(k)(1)]^(T)  (3)

The zeros insertion and repetition operation is performed at blocks 18in FIG. 1. Before transmission, the vectors v_(k,o) are multiplied by aphase shift matrix Φ_(4N) using multipliers 20:

$\begin{matrix}{{\Phi_{4N} = {{diag}\left( {1,^{\frac{j2\pi}{4N}},^{\frac{{j2\pi}\; 2}{4N}},\ldots \mspace{11mu},^{\frac{{j2\pi}{({{4N} - 1})}}{4N}}} \right)}},} & (4)\end{matrix}$

to create one element forward shift in their Fourier transform. Thetransmitted signals vector from the two antennas are then given by

x _(k)=√{square root over (P _(o)/8N)}(v _(k,e)+Φ_(4,N) v _(k,o)),k=1,2  (5)

where P_(o) is the total transmitted power. As shown in FIG. 1, adders22 are used for the (v_(k,e)+Φ_(4,N)v_(k,o)) operation in (5). At blocks24, a cyclic prefix (CP) is added to each block before transmittingthrough a frequency selective fading channel of order L.h_(k)=[h_(k)(0),h_(k)(1), . . . , h_(k)(L−1)] where h_(k)(l) is thel^(th) response of the channel impulse response (CIR) between the k^(th)transmit antenna and the receive antenna. The received signal vector yis given by

$\begin{matrix}{y = {{\sum\limits_{k = 1}^{2}{x_{k}^{({CP})} \otimes h_{k}}} + w}} & (6)\end{matrix}$

where x_(k) ^((CP)) is x_(k) with the CP added and w is an added noisevector assumed AWGN with variance N_(o) and {circle around (X)} denotesthe linear convolution. Since the CP turns the linear convolution into acircular, the received signal vector after the removal of the CP andtaking the 4N-point DFT is given by

$\begin{matrix}{Y = {{\sum\limits_{k = 1}^{2}{\Lambda_{k}X_{k}}} + W}} & (7)\end{matrix}$

where Λ_(k), k=1, 2 represent diagonal matrices whose elements are the4N-point DFT of the corresponding CIR h_(k)·X_(k) and W represents the4N-point DFT of x_(k) and w respectively.

From (5), it follows:

X _(k)=√{square root over (P _(o)/8N)}(V _(k,e) +{tilde over (V)}_(k,o)), k=1.2  (8)

where V_(k,e), V_(k,o), and {tilde over (V)}_(k,o) represent the4N-point DFT of v_(k,e), v_(k,o) and (Φ_(4N)v_(k,o)) respectively. Nowit follows that

$\begin{matrix}{{V_{k,e} = \left\lbrack {S_{k,e}^{T} \cdot S_{k,e}^{T}} \right\rbrack^{T}}{{\overset{\sim}{V}}_{2.o} = {\Phi_{4N}\left\lbrack {S_{l,o}^{T} \cdot S_{1,o}^{T}} \right\rbrack}^{T}}{{\overset{\sim}{V}}_{1.0} = {\Phi_{4N}\left\lbrack {S_{2.o}^{T} \cdot {- S_{2.o}^{T}}} \right\rbrack}^{T}}{where}} & (9) \\{{S_{k,e} = \left\lbrack {{S_{k}(0)},0,{{S_{k}(1)}0},\ldots \mspace{11mu},{S_{k^{\prime}}\left( {N - 1} \right)},0} \right\rbrack^{T}}{S_{k,o} = \left\lbrack {0,{S_{k}^{*}(0)},0,{S_{k}^{*}\left( {N - 1} \right)},\ldots \mspace{11mu},0,{S_{k}^{*}(1)}} \right\rbrack^{T}}{\overset{\sim}{\Phi}}_{4N} = {{diag}\left( {0,1,{e\frac{{jo}*}{4N}},{e\frac{{jo}*o}{4N}},\ldots \mspace{11mu},{e\frac{{jo}*\left( {{4N} - 2} \right)}{4N}}} \right)}} & (10)\end{matrix}$

It is proposed that the 4N -point DFT of a 4N elements vector of theform [p_(e) ^(T),p_(e) ^(T)]^(T) where p_(e)=[p(0),0, . . . ,p(N−1),0]^(T) is [p_(e) ^(T),p_(e) ^(T)]^(T) where P_(e)=[P(0),0, . . ., P(N−1),0]^(T) and the vector [P(0),P(1), . . . , P(N−1)]^(T) is theN-point DFT of [p(0),p(1), . . . , p(N−1)]^(T). To prove (9), it can beseen from the foregoing proposition that V_(k,e) is the 4N-point DFT ofv_(k,e)=[s_(k,e) ^(T),s_(k,e) ^(T)]^(T) where s_(k,e) is defined in (3)hence V_(k,e)=[s_(k,e) ^(T),S_(k,e) ^(T)]^(T). Next the vector {rightarrow over (s)}_(k)=[s*_(k)(0),0,s*_(k)(N−1), . . . ,0,s*_(k)(2),0,s*_(k)(1),0]^(T) is defined which is one element circularshift to the left of s_(k,o) in (3), that is s_(k,o)(n)={right arrowover (s)}_(k)(n−1) and {right arrow over (v)}_(2,o)=[{right arrow over(s)}₁ ^(T),{right arrow over (s)}₁ ^(T)]^(T) i.e. v_(2,o)(n)={rightarrow over (v)}_(2,o)(n−1).

Using the shift property of DFT, the 4N-point DFT of v_(2,o) is given by

V_(2,o)=Φ_(4N) V _(2,o)  (11)

where V _(2,o) is the 4N-point DFT of V _(2,o). Using the statementabove and applying the N-point DFT of [s*₁(0),s*₁(N−1), . . . ,s*₁(2),s*₁(1)]^(T) as [S*₁(0),S*₁(1), . . . , S*₁(N−2),S*₁)N−1)]^(T) onv _(2,o) it follows that V _(2,o)=[ S ₁ ^(T), S ₁ ^(T)]^(T) where S₁=S*_(1,e). With {tilde over (v)}_(2,o)=Φ_(4N)v_(2,o) and using theinverse of the shift property of DFT, it follows that {tilde over(V)}_(2,o)(m)=V_(2,l)(m−1) i.e. {tilde over (V)}_(2,o) is one elementcircular shift to the right of V_(2,o). Using (11), it follows that{tilde over (V)}_(2,o)=Φ_(4B)[S_(1,o) ^(T),S_(1,o) ^(T)]^(T) whereΦ_(4N) is defined in (10). One can follow the same procedure to showthat {tilde over (V)}_(1,o)=Φ_(4N)[−S_(2,o) ^(T),−S_(2,o) ^(T)]^(T).

Now for a vector A, A^(e) and A^(o) are defined to be the even and oddparts of A respectively. From (9), it follows

V_(1,e) ^(e)=[S_(k) ^(T),S_(k) ^(T)]^(T), V_(k,e) ^(o)=[0_(N) ^(T),0_(N)^(T)]^(T)

{tilde over (V)} _(1,o) ^(o)=Φ_(2N) [−S ₂ ^(H) ,−S ₁₂ ^(H)]^(T), {tildeover (V)}_(2,o) ^(o)=φ_(2N)[S_(x) ^(H),S_(l) ^(H)]^(T).

V_(k,o) ^(o)=[0_(N) ^(T),0_(N) ^(T)]^(T), k=1.2  (12)

where

$\begin{matrix}{{\Phi_{2N} = {{diag}\left( {1,^{\frac{j2\pi}{4N}},^{\frac{{j2\pi}\; 2}{4N}},\ldots \mspace{11mu},^{\frac{{j2\pi}{({{2N} - 1})}}{2N}}} \right)}},} & \;\end{matrix}$

is a 2N×2N diagonal matrix whose diagonal elements are the odd diagonalelements of Φ_(4N) of ((10)) and 0_(N) is a zero vector of length N. Itthen follows that

X _(k) ^(e)=√{square root over (P _(o)/8N)}[S _(k) ^(T) ,S _(k)^(T)]^(T)

X ₁ ^(o)=√{square root over (P _(o)/8N)}Φ_(2N) [−S ₂ ^(H) ,−S ₂^(H)]^(T)

X ₂ ^(o)=√{square root over (P _(o)/8N)}Φ_(2N) [S ₁ ^(H) ,S ₁^(S)]^(T)  (13)

From ((13)) it can be seen that:

X₂ ^(o)=Φ_(2N)X₁ ^(e)*

X ₁ ^(o)=−Φ_(2N) X ^(e)*  (14)

Eq ((7)) can then be rewritten as

$\begin{matrix}{{Y^{e} = {{\Lambda_{1}^{e}X_{1}^{e}} + {\Lambda_{2}^{e}X_{2}^{e}} + W^{e}}}\begin{matrix}{Y^{o} = {{\Lambda_{1}^{o}X_{1}^{o}} + {\Lambda_{2}^{o}X_{2}^{o}} + W^{o}}} \\{{= {{{- \Lambda_{1}^{o}}\Phi_{2N}X_{2}^{e*}} + {\Lambda_{2}^{o}\Phi_{2N}X_{1}^{e*}} + W^{o}}},}\end{matrix}} & (15)\end{matrix}$

where Λ_(k) ^(e) and Λ_(k) ^(o) are diagonal matrices whose diagonalelements are the even and odd diagonal elements of Λ_(k) respectively.Assuming that the channel gains for adjacent subcarriers areapproximately equal, i.e. Λ_(k) ^(e)≈Λ_(k) ^(o), k=1,2; hence combining((15)) gives

$\begin{matrix}{\begin{pmatrix}Y^{e} \\Y^{o*}\end{pmatrix} = {{\begin{pmatrix}\Lambda_{1}^{e} & \Lambda_{2}^{e} \\{\Phi_{2N}^{*}\Lambda_{2}^{e*}} & {{- \Phi_{2N}^{*}}\Lambda_{1}^{e*}}\end{pmatrix}\begin{pmatrix}X_{1}^{e} \\X_{2}^{e}\end{pmatrix}} + \begin{pmatrix}W^{e} \\W^{o*}\end{pmatrix}}} & (16)\end{matrix}$

The first and second N terms of Y^(e) and Y^(o) are defined respectivelyfor k=1, 2 by

Y _(k) ^(e) =[Y ^(e)((k−1)N), . . . , Y ^(e)(kN−1)]^(T)

Y _(k) ^(o) =[Y ⁰((k−1)N), . . . , Y ^(o)(kN−1)]^(T)  (17)

Plugging ((13)) and ((17)) in ((16)) arrives at:

$\begin{matrix}{{Z = {{AS} + \overset{\sim}{W}}}{where}} & (18) \\{{{Z = \left\lbrack {Y_{1}^{e^{T}},Y_{2}^{e^{T}},Y_{1}^{o*^{T}},Y_{2}^{o*^{T}}} \right\rbrack^{T}},{S = \left\lbrack {S_{1}^{T},S_{2}^{T}} \right\rbrack^{T}}}{\overset{\sim}{W} = \left\lbrack {W_{1}^{e^{T}},W_{2}^{e^{T}},W_{1}^{o^{T}},W_{2}^{o}} \right\rbrack^{T}}{\Lambda = {\sqrt{{P_{o}/8}N}\begin{pmatrix}\begin{matrix}\Lambda_{1,1}^{e} & \Lambda_{2,1}^{e} \\\Lambda_{1,2}^{e} & \Lambda_{2,2}^{e}\end{matrix} \\{{\Phi_{{2N},1}^{*}\Lambda_{2,1}^{e*}} - {\Phi_{{2N},1}^{*}\Lambda_{1,1}^{e*}}} \\{{\Phi_{{2N},2}^{*}\Lambda_{2,2}^{e*}} - {\Phi_{{2N},2}^{*}\Lambda_{1.2}^{e*}}}\end{pmatrix}}}} & (19)\end{matrix}$

where Λ_(k,1) ^(e) and Λ_(k,2) ^(e) are N×N diagonal matrices whosediagonal elements are the first and last N diagonal elements of Λ_(k)^(e) and Λ_(2,1) ^(e), Λ_(2,2) ^(e), Φ_(2N,1), Φ_(2N,2) are similarlydefined. Note that |Φ_(2N)|²=I_(2N),|Φ_(2N,1)|²=|Φ_(2N,2)|²=I_(N) wherefor a diagonal matrix D we defined |D|²=DD*. The proposedspace-frequency decoder gives the estimation Ŝ according to thefollowing

{circumflex over (S)}=(Λ^(H)Λ)⁻¹Λ^(H) Z  (20)

It can be shown that the matrix Λ^(H)Λ is diagonal and given by

$\begin{matrix}{{\Lambda^{H}\Lambda} = \begin{pmatrix}{\Delta }^{2} & 0 \\0 & {\Delta }^{2}\end{pmatrix}} & (21)\end{matrix}$

where

${\Delta }^{2} = {\frac{P_{o}}{8N}\left( {{\Lambda_{1,1}^{e}}^{2} + {\Lambda_{1,2}^{e}}^{2} + {\Lambda_{2,1}^{e}}^{2} + {\Lambda_{2,2}^{e}}^{2}} \right)}$

hence S₁ and S₂ are completely decoupled. The estimates in (20) aretransformed back in time domain for detection.

FIG. 2 illustrates operations performed in an exemplary single carrierspace-frequency (SF-SCFDE) transmission technique utilized bytransmitter 10 described with reference to FIG. 1. Additional, fewer ordifferent operations may be performed depending on the embodiment. In anoperation 30, data symbols d(n) belonging to an alphabet A aredemultiplexed into two block streams of N symbols. In an operation 32,the block streams are encoded into communication blocks.

The communication blocks are compressed into vectors in an operation 34and the compressed vectors are repeated twice to form vector blocks inan operation 36. Once the vector blocks are formed, they are phaseshifted using a phase shift matrix (operation 38). This phase shiftcreates a one element forward shift in the Fourier transform. In anoperation 40, a cyclic prefix (CP) is added to each phase shifted vectorblock. Once the CP prefix is added, the blocks are transmitted through afrequency selective fading channel of order L. Upon receipt of thecommunicated blocks, the CP prefix is removed and a 4-N point DFT isdetermined.

The exemplary single carrier space-frequency (SF-SCFDE) transmissionscheme over frequency selective and fast fading channel described hereinhas been shown to be an efficient and effective transmission techniqueespecially for application where channel is fast fading. The bit errorrate (BER) performance of the exemplary space-frequency single carriersystem was calculated in a simulation. The simulation used a singlecarrier transmission with N=64 data symbols per block in a frequencyselective channel assumed to be a COST207 six-ray (L=6) typical urbanchannel. The BER performance was shown to outperform the ST-OFDMdescribed by K. F. Lee and D. B. Williams in “A space-time codedtransmitter diversity technique for frequency selective fading,” in IEEESensor Array and Multichannel Signal Processing Workshop, pp. 149-152,March 2000 and “A space-frequency diversity technique for OFDM system,”IEEE GLOBECOM, pp. 1473-1477, November 2000 (referred to below as “Leeand Williams”). It also outperformed the conventional OFDM system in thesame channel. The performance of SF-SCFDE described herein was alsocompared with that of ST-SCFDE described in W. M. Younis, N. Al-Dhahir,and A. H. Sayed, “Adaptive frequency-domain equalization of space-timeblock-coded transmissions,” in IEEE Int. Conf. Accoust., Speech, SignalProcess., vol. 3, Orlando, Fla. May 2002, pp. 2353-2356 (referred tobelow as “Younis”), in slow fading channel (where the normalized Dopplerfrequency is 0.001) and fast fading channel (where the normalizedDoppler frequency is 0.05). Simulation results show that the SF-SCFDEscheme described herein depicts much better BER. One reason for thebetter performance is frequency domain spreading which causes additionalfrequency domain diversity. Furthermore, the techniques of the exemplaryembodiments do not suffer the PAPR (peak to average power ratio)problem.

FIG. 3 is a graph showing a performance comparison of the exemplarySF-SCFDE described herein with ST-SCFDE of Lee and Williams (left) andSF-SCFDE with SF-OFDM of Younis (right) in slow (fdTs=0.001) and fast(fdTs=0.05) fading COST207 six-ray typical urban (TU) channel.

The foregoing description of exemplary embodiments has been presentedfor purposes of illustration and description. It is not intended to beexhaustive or to limit the present invention to the precise formdisclosed, and modifications and variations are possible in light of theabove teachings or may be acquired from practice of the presentinvention. The embodiments were chosen and described in order to explainthe principles of the present invention and its practical application toenable one skilled in the art to utilize the present invention invarious embodiments and with various modifications as are suited to theparticular use contemplated.

1. A method for single carrier space-frequency (SF-SCFDE) transmissionover frequency selective and fast fading channel, the method comprising:receiving communication block streams; encoding the receivedcommunication block streams to produce communication blocks; adding acyclic prefix to each communication block to form transmission blocks;and communicating each transmission block through a frequency selectivefading channel.
 2. The method of claim 1, wherein receivingcommunication block streams further comprises demultiplexing datasymbols into multiple communication block streams.
 3. The method ofclaim 1, wherein encoding the received communication block streams toproduce communication blocks further comprises: compressing thecommunication blocks into compressed vectors; repeating the compressedvectors twice to form vector blocks; and phase shifting the vectorblocks using a phase shifting matrix.
 4. The method of claim 3, whereinthe phase shifting matrix is Φ_(4N) and is defined by:$\Phi_{4N} = {{{diag}\left( {1,^{\frac{j2\pi}{4N}},^{\frac{{j2\pi}\; 2}{4N}},\ldots \mspace{11mu},^{\frac{{j2\pi}{({{4N} - 1})}}{4N}}} \right)}.}$5. The method of claim 1, wherein the frequency selective fading channelis of an order L such that a channel impulse response between a transmitantenna and a receive antenna is represented ash_(k)=[h_(k)(0),h_(k)(1), . . . , h_(k)(L−1)] where h_(k)(l) is thel^(th) response.
 6. The method of claim 1, wherein communicating eachtransmission block through a frequency selective fading channelcomprises transmitting the transmission block by a k^(th) transmitantenna using an N-point Discrete Fourier Transform (DFT) where a4N-point DFT of a 4N elements vector of the form [p_(e) ^(T), p_(e)^(T)]^(T) where p_(e)=[p(0),0, . . . , p(N−1),0]^(T) is [p_(e)^(T),p_(e) ^(T)]^(T) where P_(e)=[P(0),0, . . . , P(N−1),0]^(T) and thevector [P(0), P(1), . . . , P(N−1)]^(T) is the N-point DFT of [p(0),p(1), . . . , p(N−1)]^(T).
 7. The system of claim 1, further comprising,at a receive antenna, removing the cyclic prefix and taking a 4N-pointDiscrete Fourier Transform (DFT).
 8. An apparatus for single carrierspace-frequency (SF-SCFDE) transmission over frequency selective andfast fading channel, the apparatus comprising: a space-frequency encoderthat receives and encodes communication block streams; a cyclic prefixadder that adds a cyclic prefix to each communication block; andantennas that communicate each communication block.
 9. The apparatus ofclaim 8, wherein the space-frequency encoder further comprises acompressor to compress the encoded communication block streams,circuitry to repeat the communication block streams and form vectorblocks, and a phase shifter to perform a phase shift.
 10. The apparatusof claim 9, wherein the phase shift is done using a phase shiftingmatrix, Φ_(4N), defined by:$\Phi_{4N} = {{{diag}\left( {1,^{\frac{j2\pi}{4N}},^{\frac{{j2\pi}\; 2}{4N}},\ldots \mspace{11mu},^{\frac{{j2\pi}{({{4N} - 1})}}{4N}}} \right)}.}$11. The apparatus of claim 8, further comprising a demultiplexer thatdemultiplexes data symbols into multiple communication block streams.12. The apparatus of claim 8, wherein the antennas that communicate eachcommunication block communicate using a fast fading channel.
 13. Asystem for single carrier space-frequency (SF-SCFDE) transmission overfrequency selective and fast fading channel, the system comprising: atransmitter having a space-frequency encoder that receives and encodescommunication block streams, a cyclic prefix adder that adds a cyclicprefix to each communication block, and antennas that communicate eachcommunication block; and a receiver having antennas that receivecommunication blocks from the transmitter, a structure to remove thecyclic prefix, and a decoder.
 14. The system of claim 13, wherein thespace-frequency encoder further comprises a compressor to compress theencoded communication block streams, circuitry to repeat thecommunication block streams and form vector blocks, and a phase shifterto perform a phase shift.
 15. The system of claim 14, wherein the phaseshift uses a phase shifting matrix, Φ_(4N), defined by:$\Phi_{4N} = {{{diag}\left( {1,^{\frac{j2\pi}{4N}},^{\frac{{j2\pi}\; 2}{4N}},\ldots \mspace{11mu},^{\frac{{j2\pi}{({{4N} - 1})}}{4N}}} \right)}.}$16. The system of claim 13, wherein the transmitter further includes ademultiplexer that demultiplexes data symbols into multiplecommunication block streams.
 17. The system of claim 13, whereinantennas communicate the communication blocks through a frequencyselective fading channel.
 18. The system of claim 17, wherein thefrequency selective fading channel is of an order L such that a channelimpulse response between the transmitter and the receiver is representedas h_(k)=[h_(k)(0), h_(k)(1), . . . , h_(k)(L−1)] where h_(k)(l) is thel^(th) response.
 19. The system of claim 17, wherein communicationthrough the frequency selective fading channel uses a k^(th) transmitantenna with an N-point Discrete Fourier Transform (DFT) where a4N-point DFT of a 4N elements vector of the form [p_(e) ^(T), p_(e)^(T)]^(T) where p_(e)=[p(0),0, . . . , p(N−1),0]^(T) is [P_(e)^(T),P_(e) ^(T)]^(T) where P_(e)=[P(0),0, . . . , P(N−1),0]^(T) and thevector [P(0),P(1), . . . , P(N−1)]^(T) is the N-point DFT of [p(0),p(1), . . . , p(N−1)]^(T).
 20. The system of claim 16, wherein thereceiver takes a 4N-point Discrete Fourier Transform (DFT) of thecommunication blocks after the cyclic prefix is removed.